Question

Find the number of permutations of all the letters of the word “MATHEMATICS” which starts with consonants only.

Answer 1

To find the number of permutation of consonants starting word, we separate the consonants and then find permutation of each of those consonants. Using the permutation formula for each letter group is:
$P({{n}_{1}},{{n}_{2}},...n)=\dfrac{X!}{{{n}_{1}}!,{{n}_{2}}!,...n!}$
where ${{n}_{1}},{{n}_{2}},...n$ are the group of consonants and $X$ is the total number letters in the word given.

Complete step-by-step answer:
The total number of letters in the word “MATHEMATICS” is $10$.
Now separating the letter groups we have:
For T starting as first letter in the word:
$P(T)=\dfrac{10!}{2!.2!}$
For M starting as first letter in the word:
$P(M)=\dfrac{10!}{2!.2!}$
For H, C, S starting as first letter in the word:
$P(H,C,S)=3\times \dfrac{10!}{2!.2!.2!}$
Hence, the total possibility of permutation is given as:
$P(T,T,M,M,H,C,S)=3\times \dfrac{10!}{2!.2!.2!}+\dfrac{10!}{2!.2!}+\dfrac{10!}{2!.2!}$
$=3\times \dfrac{10!}{2!.2!.2!}+2\times \dfrac{10!}{2!.2!}$
$=3\times \dfrac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1.2\times 1.2\times 1}+2\times \dfrac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1.2\times 1}$
$=3\times \dfrac{10\times 9\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{1}+\dfrac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 1}{1}$
$=3175200$

Hence, the number of ways consonants starting the word is $3175200$.

Note: Students may be wrong while solving the sum of the factorial value of multiple groups that are solved above. Another way to solve the sum easily is to find commons in the denominator:
$=3\times \dfrac{10!}{2!.2!.2!}+2\times \dfrac{10!}{2!.2!}$
$=\dfrac{10!}{2!.2!}\left[ 3\times \dfrac{1}{2!}+2\times \dfrac{1}{1} \right]$
$=\dfrac{10!}{2!.2!}\left[ \dfrac{3}{2}+2 \right]$
$=\dfrac{10!}{2!.2!}\left[ \dfrac{7}{2} \right]$
$=3175200$